Birkhoff-James classification of norm's properties

Abstract

For an arbitrary normed space $\mathcal X$ over a field $\mathbb F \in \{ \mathbb R, \mathbb C \}$, we define the directed graph $Γ(\mathcal X)$ induced by Birkhoff-James orthogonality on the projective space $\mathbb P(\mathcal X)$, and also its nonprojective counterpart $Γ_0(\mathcal X)$. We show that, in finite-dimensional normed spaces, $Γ(\mathcal X)$ carries all the information about the dimension, smooth points, and norm's maximal faces. It also allows to determine whether the norm is a supremum norm or not, and thus classifies finite-dimensional abelian $C^\ast$-algebras among other normed spaces. We further establish the necessary and sufficient conditions under which the graph $Γ_0(\mathcal{R})$ of a (real or complex) Radon plane $\mathcal{R}$ is isomorphic to the graph $Γ_0(\mathbb F^2, \|\cdot\|_2)$ of the two-dimensional Hilbert space and construct examples of such nonsmooth Radon planes.

Figures (2)

• Figure 1: The original norm on $\mathbb{R}^2$.
• Figure 2: Absolute nonsmooth Radon plane.

Theorems & Definitions (62)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Corollary 2.3
• proof